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Using undrained triaxial shear tests, this study investigates the mechanical properties of fine-grained materials (silty clay and sand) which are extensively used for China's Qinghai–Tibet Railway (QTR) under different confining pressures (σ3) and freezing temperatures (T). The results show that a reduction in T causes an increase in the shear strength and elastic modulus of all the materials tested in the present study. In addition, the freezing of the silty clay has no significant effect on the type of soil behaviour (strain-hardening), whereas the freezing of the sand changes its strain-hardening behaviour to strain-softening. Supposing that the deviatoric stress–strain curves of the silty clay and sand can be divided into two segments due to a reverse bending point, it was assumed that the first segment follows a hyperbolic function. Meanwhile, the second segment is also a hyperbola, with the reverse bending point as the origin and the residual strength as the asymptote. Accordingly, a nonlinear relation constitutive model that considers σ3 and T is derived. All model parameters are identified. The reasonability of the new model was verified using the test results of the materials. A comparison of the predicted and test results shows that this model can well simulate the deviatoric stress–strain response in the failure process of the tested materials. In particular, it can reflect the residual deviatoric stress after the materials' failure. [ABSTRACT FROM AUTHOR]